One problem with Mangled World and similar explanations is that it just shows that the Born probabilities are actually a result of another true Born probability. For example, if some of those planes in that story get destroyed so that the probability of one surviving is proportional to the square of its thickness, you now have to wonder why you're equally likely to be in each of the worlds.

Don't get me wrong. Being equally likely makes a lot more sense than the likelihood being proportional to the square of the thickness, but the likelihood being proportional to the thickness would also make some sense.